Title: 3D Printed Cross Sections of the Menger Sponge Post by: tit_toinou on April 02, 2013, 10:03:15 AM Hi all,
I made the cross sections of the menger sponge on Shapeways.com : (http://images1.sw-cdn.net/model/picture/674x501_1006153_910614_1364824047.jpg) Étoile de Menger 8cm (http://www.shapeways.com/model/1006153/etoile-de-menger.html) (http://images1.sw-cdn.net/model/picture/674x501_1006154_910615_1364824079.jpg) Étoile de Menger 5cm (http://www.shapeways.com/model/1006154/etoile-de-menger-petite.html) (http://images1.sw-cdn.net/model/picture/674x501_1006224_910698_1364827157.jpg) Triangle de Menger (http://www.shapeways.com/model/1006224/triangle-de-menger.html) (http://images1.sw-cdn.net/model/picture/674x501_1006218_910694_1364826894.jpg) Losange de Menger (http://www.shapeways.com/model/1006218/losange-de-menger.html) Title: Re: 3D Printed Cross Sections of the Menger Sponge Post by: cKleinhuis on April 02, 2013, 10:46:46 AM that star shapes are just awesome, as i told before, i like that they are just inside the menger when cutting diagonally,
i wonder how smooth the diagonal parts are, because the rasterization may lead to pixel-stepping !? Title: Re: 3D Printed Cross Sections of the Menger Sponge Post by: hobold on April 02, 2013, 07:33:09 PM By the way, there is at least one more interesting shape hiding in that Menger sponge.
Starting from the whole cube, you can cut away a total of four "Triangle de Menger" pieces (2nd to last picture). You end up with a regular tetrahedron that has four identical faces, each looking like the equilateral face of of a "Triangle de Menger". Title: Re: 3D Printed Cross Sections of the Menger Sponge Post by: tit_toinou on April 02, 2013, 11:02:00 PM that star shapes are just awesome, as i told before, i like that they are just inside the menger when cutting diagonally, I'm mesmerised too.i wonder how smooth the diagonal parts are, because the rasterization may lead to pixel-stepping !? There's no pixel-stepping. I didn't do (for example) what bib does for creating 3d models, i.e. creating little cubes from a DE formula. I created individually all the little cubes. And I computed the intersections for the cubes crossing the plane. Computing cross sections of a cube is a pain in the ass, by the way ^^ . By the way, there is at least one more interesting shape hiding in that Menger sponge. How do you "cut" a cube with fours planes and end up with a tetrahedron ?Starting from the whole cube, you can cut away a total of four "Triangle de Menger" pieces (2nd to last picture). You end up with a regular tetrahedron that has four identical faces, each looking like the equilateral face of of a "Triangle de Menger". I can see how to cut it with height planes and end up with an octahedron ! I'll try to create it I think it'll be worth it. I've also spotted a mix between "Losange de Menger" and "Triangle de Menger" I have to render it. Title: Re: 3D Printed Cross Sections of the Menger Sponge Post by: hobold on April 04, 2013, 04:29:42 PM How do you "cut" a cube with fours planes and end up with a tetrahedron ? Consider the eight corners of a cube. Imagine coloring those corners with a checkerboard pattern, that is, pick one corner and color it black. Then color the three adjacent corners white. Now the immediate neighbors of the new white corners can be colored black, and so on. Eventually you will have four black corners and four white corners.Now you can cut four tetrahedra from the original cube in the following way: start from a black corner and place a cutting plane through three adjacent white corners. This can be done for all four black cube corners. Title: Re: 3D Printed Cross Sections of the Menger Sponge Post by: tit_toinou on April 05, 2013, 01:54:24 PM @hobold: Ok i'll try that.
I've found a way to get three designs into one object ^^ : (http://images1.sw-cdn.net/model/picture/674x501_1012317_917857_1365088172.jpg) Tranches de Menger (http://www.shapeways.com/model/1012317/tranches-de-menger.html) . Title: Re: 3D Printed Cross Sections of the Menger Sponge Post by: cKleinhuis on April 29, 2013, 03:35:56 PM just for completion, haltenny posted this in response the menger contact https://simonsfoundation.org/multimedia/mathematical-impressions-the-surprising-menger-sponge-slice/ |